Libor Market Mode - Theory and Practice

Inhaltsverzeichnis (Table of Contents)

• 1. Introduction
• 2. Comparison with Alternative Interest Rate Models
• 3. General Option Pricing
• 3.1. Fundamentals of Derivatives Valuation
• 3.2. Change-of-Numeraire Theorem
• 3.3. Girsanov's Theorem
• 4. Libor Market Model Theory: Arbitrage-free Forward Libor Rate Dynamics
• 4.1. Forward Libor Rate Process as a Martingale
• 4.2. Dynamics of Forward Libor Rates under the Forward Measure
• 4.3. Extension to Several Factors
• 5. Obtaining the Data Input for the Libor Market Model
• 5.1. Derivation of Time Zero Forward Libor Rates
• 5.2. Calibration of Volatility Parameters to Cap Prices
• 5.3. Calibration to Swaption Prices
• 5.3.1. Calibration of Correlation and Volatility Parameters to Swaptions
• 5.3.2. Cascade Calibration
• 6. Forward Libor Rates Volatility Modeling
• 6.1. The Term Structure of Volatility
• 6.2. Constant Volatility Structure
• 6.3. Piecewise-Constant Volatility Structure
• 6.4. Parametric Volatility Structure
• 6.5. Determining of Volatility Parameters with the Two-step Approach
• 7. Forward Libor Rates Correlation Modeling
• 7.1. Specifications of Forward Rate Correlation
• 7.1.1. Full Rank Specification with Reduced Number of Parameters
• 7.1.2. Reduced-Rank Correlation Specifications
• 7.2. Obtaining an Exogenous Correlation Matrix for Cascade Calibration
• 7.2.1. Step 1: Historical Estimation of Correlation Matrix
• 7.2.2. Step 2 and 3: Fitting Historically Estimated Correlation Matrix to a Parametric Form and Reducing the Rank
• 8. Hedging
• 9. The Libor Market Modell: Practice
• 9.1. Implementation Steps with Monte Carlo Simulations
• 9.2. Implementation of the LMM: Results
• 9.2.1. Study 1: Valuation of Caplets and Caps
• 9.2.2. Study 2: Valuation of Discrete Barrier Caps
• 9.2.3. Study 3: Cascade Calibration
• 9.2.4. Study 4: Valuation of European Swaptions
• 9.2.5. Study 5: Valuation of Ratchets

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis aims to explore the theoretical foundation and practical application of the Libor Market Model (LMM) for pricing and hedging interest rate derivatives. The work focuses on providing a comprehensive understanding of the model's mechanics, calibration techniques, and implementation for various financial instruments.

• Forward Libor Rate Dynamics
• Calibration Techniques for Volatility and Correlation Parameters
• Model Implementation for Pricing and Hedging
• Practical Applications of the LMM for various Derivatives
• Comparison with Alternative Interest Rate Models

Zusammenfassung der Kapitel (Chapter Summaries)

The thesis commences with an introduction to the Libor Market Model, outlining its significance in modern finance and presenting an overview of the research conducted. Chapter 2 delves into a comparison of the LMM with other prominent interest rate models, highlighting their strengths and limitations. Chapter 3 provides a general overview of option pricing theory, establishing the foundation for understanding the model's valuation framework. This chapter covers fundamental principles of derivatives valuation, the change-of-numeraire theorem, and Girsanov's theorem.

Chapter 4 dives into the heart of the LMM, exploring its theoretical basis. This chapter examines the arbitrage-free dynamics of forward Libor rates, demonstrating how these rates can be modeled as martingales. The chapter also discusses the dynamics of forward Libor rates under the forward measure and explores the extension of the model to incorporate multiple factors.

Chapter 5 focuses on obtaining the necessary data inputs for the LMM, including the derivation of time zero forward Libor rates and the calibration of volatility parameters to cap prices. The chapter further details the calibration process for swaption prices, including the application of both correlation and volatility parameters.

Chapter 6 delves into the modeling of forward Libor rates volatility. The chapter explores various volatility structures, including constant volatility, piecewise-constant volatility, and parametric volatility specifications. It also discusses the determination of volatility parameters using the two-step approach.

Chapter 7 examines the modeling of forward Libor rates correlation. This chapter outlines different specifications for forward rate correlation, including full-rank and reduced-rank specifications. It further explains the process of obtaining an exogenous correlation matrix for cascade calibration, which involves historical estimation, fitting to a parametric form, and reducing the rank.

Chapter 8 explores hedging strategies within the context of the LMM. This chapter focuses on the application of the model to hedging various interest rate derivatives.

Chapter 9 delves into the practical implementation of the LMM. This chapter details the implementation steps using Monte Carlo simulations and presents the results of several studies. These studies include the valuation of caplets, caps, discrete barrier caps, European swaptions, and ratchets.

Schlüsselwörter (Keywords)

The key focus of this thesis is on the Libor Market Model (LMM), an important framework for pricing and hedging interest rate derivatives. The study examines various aspects of this model, including its theoretical foundation, calibration techniques, and practical implementation. The work highlights key topics such as forward Libor rate dynamics, volatility and correlation modeling, and Monte Carlo simulations. It explores the application of the LMM for valuing and hedging various interest rate derivatives, such as caps, floors, swaptions, and barrier options. This research uses concepts such as martingales, change-of-numeraire, and Girsanov's theorem, which are crucial for understanding the model's mathematical underpinnings.